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\title{\vspace{-4cm}\textbf{河北师范大学数学分析真题}}
\author{宁鑫雨}
\date{\today}
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\date{}
\section*{2014年数学分析}
\begin{problem}[本题30分,每题10分]\\
    1)求极限$\displaystyle\lim_{x\to0}{\frac{1}{\sin^{4}x}}\int_{0}^{x^{2}}\tan t\ d t$\\
    2)求积分$\int_{0}^{\frac{\pi}{2}}\frac{\cos\theta}{1+\sin^{2}\theta}\d\theta $\\
    3)计算曲线积分$\int_{L}\frac{-y}{x^{2}+y^{2}}\ d x+\frac{x}{x^{2}+y^{2}}\ d y$,其中$L$为曲线$y=x^{2}-1$自$A(-1,0)$至$B(2,3)$的弧段.\\
\end{problem}

\begin{problem}[本题10分]
设函数$f(x)$在$(0,+\infty)$上有定义,$f(x^{2})=f(x)$,且$\displaystyle\lim_{x\to0^{+}}f(x)=\displaystyle\lim_{x\to+\infty}f(x)=f(1)$,
证明:$f(x)$为常量函数，$x\in (0,+\infty)$.\\
\end{problem}

\begin{problem}[本题10分]
设函数$f(x)$在$[0,+\infty)$上可微,且$0\leq f^{\prime}(x)\leq f(x)$,$f(0)=0$,
证明：在$[0,+\infty)$上$f(x)\equiv0$.
\end{problem}

\begin{problem}[本题10分]
    已知曲线$y=f(x)$在$x=0$处的切线方程为$y=10x$,令
    $g(x)=\left\{
        \begin{matrix}
            {{\frac{1}{x}\int_{0}^{1}f(x^{2}t)\d t,\quad x\neq0;}}\\
            {0,\quad x=0.}\\
        \end{matrix}
        \right.$ 求$g^{\prime}(0)$.
\end{problem}

\begin{problem}[本题15分]
设函数$f(x)=\ln x+{\frac{1}{x}}\quad(x>0)$,\\
1)求$f(x)$的最小值;\\
2)设数列$\{x_{n}\}$满足$\ln x_{n}+\frac{1}{x_{n+1}}<1$,证明:$\displaystyle\lim_{n\to\infty}x_n$存在,并求此极限.
\end{problem}

\begin{problem}[本题15分]
    设函数$f(x)$在$[2,5]$上连续可微,且$f(2)=0$,证明:$\int_{2}^{5}(f^{\prime}(x))^{2}dx\geq\frac{M^{2}}{3}$,其中$M=\displaystyle\sup_{x\in[2,5]}f(x)$.
\end{problem}

\begin{problem}[本题15分]
    设函数$f(x)$在$[0,1]$上连续,在$(0,1)$可导,且$f(0)=f(1)=0$,$f(\frac{1}{2})=1$.证明：\\
    1)存在$\eta\in\left({\frac{1}{2}},1\right)$,使得$f(\eta)=\eta $;\\
    2)对任意实数$\lambda$,一定存在$\xi \in (0,\eta)$,使得$f^{\prime}(\xi)-\lambda(f(\xi)-\xi)=1$.
\end{problem}

\begin{problem}[本题15分]
    设$f(x)$为$(0,+\infty)$上连续减函数,$f(x)>0$,又设$a_{n}=\displaystyle\sum_{k=1}^{n}f\left(k\right)-\int_{1}^{n}f\left(x\right)\ d x$,证明${a_{n}}$为收敛数列.
\end{problem} 

\begin{problem}[本题15分]
    设求幂函数$\displaystyle\sum_{n=1}^{\infty}\frac{n}{n+1}(x-2)^{n+1}$的收敛域及和函数.
\end{problem}

\begin{problem}[本题15分]
    设$W=W(x,y)$二阶可微,在$u=x^{2}-y^{2}$,$\nu=2xy$下，
    证明:$\frac{\partial^{2}W}{\partial x^{2}}+\frac{\partial^{2}W}{\partial y^{2}}=4(x^{2}+y^{2})(\frac{\partial^{2}W}{\partial u^{2}}+\frac{\partial^{2}W}{\partial\nu^{2}})$.
\end{problem}
\end{document}